Smallest Enclosing Circle Problem

Introduction

The minimal enclosing circle is used in planning the
location of a shared facility. For example, a shared facility is a hospital
servicing a community. Note that traditionally people consider post office as an
example hence post office problem (see Voronoi diagram). If we consider each
home in a community as points in the plane, finding minimal enclosing circle
gives a good place to put the hospital i.e., the center of the minimal circle.
Placing the hospital at the center of minimal circle minimizes the distance
between the hospital and the farthest home (point) in the community.

In the military, this problem is known as the “Bomb
Problem”. If we suppose each target on a map as a planar point, the center of
the minimal circle of a map i.e., set, is a good place to drop a bomb for
maximum destruction. Furthermore, the radius of the minimal circle can be used
to calculate how much explosive is required.

This problem is also useful to examine the point that lie
on the boundary of the minimal enclosing circle. These points are in a sense the
outliers of the set, and in statistics, are sometimes discarded to get a more
robust estimate.

The simplest algorithm considers every circle defined by
two or three of the n points, and finds the smallest of “these” circles that
contains every point. There exits O(n3) such circles, and each takes
O(n) time to check, for a total running time of O(n4). Elzinga and
Hearn gave an O(n2) algorithm in 1972, and Shamos and Hoey (1975),
Preparata (1977), and Shamos (1978) discovered the first O (n logn) algorithms.

Finally, and to everyone's surprise, in 1983 Nimrod Megiddo
showed that the minimal enclosing circle problem can be solve in O(n) time using
the prune-and-search techniques for linear programming. This landmark result is
one of the most beautiful in the field of computational geometry.

Problem Statement

The Minimal Enclosing Circle Problem is, simply stated, the
problem of finding the smallest circle that completely contains a set of points.
Formally, given a set S of n planar points, find the circle C of smallest radius
such that all points in S are contained in either C or its boundary.

An O(n2)-time Algorithm

Draw a circle at center, c, such that points of given set lie within
the circle. Clearly, this circle can be made smaller (or else we have the
solution).

Make a circle smaller by finding the point A farthest from the center of
circler, and drawing a new circle with the same center and passing through the
point A. These two steps produce a smaller enclosing circle. The reason that the
new circle is smaller is that this new circle still contains all the points of
given set, except now it passes through farthest point, x, rather than enclosing
it.

If the circle passes through 2 or more points, proceed to step 4. Otherwise,
make the circle smaller by moving the center towards point A, until the circle
makes contact with another point B from the set.

At this stage, we a circle, C, that passes through two or more points of a
given set. If the circle contains an interval (point-free interval) of
arc greater than half the circle's circumference on which no points lie, the
circle can be made smaller. Let D and E be the points at the ends of this
point-free interval. While keeping D and E on the circle's boundary, reduce the
diameter of the circle until we have either case (a) or case (b).

Case (a) The diameter is the
distance DE.

We are done!

Case (b) The circle C touches
another point from the set, F.

Check whether there exits
point-free arc intervals of length more than half the circumference of C.

IF no such point-free arc
intervals exit THEN

We are done!

Else

Goto step 4.

In this case,
three points must lie on an arc less than half the circumference in length. We
repeat step 4 on the outer two of the three points on the arc.

Analysis

This algorithm is straight forward, but expensive to
implement. Step 1, 2 and 3 require linear time in the number of points in the given set.
In step 4 above, it takes linear time to find each new point F. However, finding
the point F does not guarantee the termination of the algorithm. Step 4 must be
repeated until the circle contains no point-free interval longer than half its
circumference. In the worst case, this requires (n-2) iterations of step 4,
implying that the total time spent in step 4 could be on the order of the square
of the size of the point set.

Hence, this algorithm is O(n2).

The remaining section is divided into two parts. First part
describes the algorithm used in implementation and the second part presents the
best-known algorithm for finding minimal enclosing circle, MEC.

Applet’s Algorithm

This algorithm is due to Pr. Chrystal (1885).

The Algorithm

Foremost, observe that the minimal enclosing circle, MEC, is
entirely determined by the Convex Hull of a given set of point. The
reason is the points of the set touched by the minimal enclosing circle are
always on the convex hull of the set.

First step of the algorithm is to compute the convex hull, H, of the points in linear
time. Clearly, we can do this since points are kept ordered by x-coordinate.
Call the convex hull H and the number of convex hull vertices h. Next step is to pick any side, say S, of the convex hull.
The main loop of the algorithm is as follows:

Main Loop

For each vertex of H other than those of S, we compute
the angle subtended by S. The minimum such angle occurs at vertex v,
and the value of the minimum angle is α (alpha).

IF α ≥ 90 degrees THEN

We are done!

/* The minimal enclosing circle is
determined by the diametric circle of S */

IF α < 90 degrees THEN

Goto step 2 below.

Since α < 90 degrees, check the remaining
vertices of the triangle formed by S and v.

IF no vertices are obtuse THEN

We are done!

/* The MEC is determined by the vertices of S and the vertex v */

If one of the other angles of the triangle formed by S
and v is obtuse, then set S to be the side opposite the obtuse angle and
repeat the main loop of the algorithm (i.e., Goto step 1 above).

Analysis

Assuming the points are given in sorted order, the
algorithm initializes in the time that is linear in the number of points in the
set. The main loop of the algorithm requires linear time with respect to the
number of convex hull points, and main loop could be repeated once for each
diagonal of the convex hull. The number of diagonals is proportional to the
square of the number of convex hull points.
For that reason in the worst case, the total time of the algorithm is
proportional to the number of points in the set, plus the cube of the
number of convex hull points.
However, in implementation, the running time depends on the side initially
chosen to start the algorithm, and the algorithm expected to perform quite well
in practice.

Halting Condition

It remains to prove that the algorithm will defines the minimal enclosing
circle of the given set of points, rather than looping forever. The proof
follows from the fact that during each iteration, algorithm is reducing the
radius of the circle being considered and ensuring that all the points of the
set are still within the circle. Therefore, the circle will eventually become
the minimal enclosing circle.

Linear Time Algorithm

Nimrod Megiddo proposes the algorithm and it uses the prune-and-search
techniques for linear programming to find the minimal enclosing circle in O(n)
time.

Prune-and-Search

The essence of Megiddo's algorithm is prune-and-search.
In a prune-and-search algorithm, a linear amount of work is done at each step to
reduce the input size by a constant fraction f. If this can be achieved, then
the total amount of work done will reduce to O(n)*(1 + (1-f) + (1-f)2 + ...).
In this formula, the infinite series is geometric and sums to a constant value,
and so the total running time is O(n). For example, suppose that by inspecting
our set of n input elements we can discard 1/4 of them as irrelevant to the
solution. By repeatedly applying this inspection to the remaining elements, we
can reduce the input to a size which is trivial to solve, say n≤3. The total
time taken to achieve this reduction will be proportional to
(n + 3n/4 + 9n/16 + ...). It is easy to show that this series approaches, and
never exceeds, a limit of 4n. Therefore, the total running time is proportional
to n, as required.

The idea of using the geometric series to reduce an algorithm to linear time
predates Megiddo's work; in particular, it had previously been used to develop
O(n) median-finding algorithms. However, he was the first to apply it to a
number of fundamental problems in computational geometry.

Megiddo's Linear-Time Algorithm

To find the minimal enclosing circle (MEC) of a set of
points, Megiddo's algorithm discards at least n/16 points at each (linear-time)
iteration. That is, given a set S of n points, the algorithm identifies n/16
points that can be removed from S without affecting the MEC of S. This procedure
can be repeatedly applied until some trivial base case is reached (such as n=3),
with the total running time proportional to (n + 15n/16 + 225n/256 + ...) = 16n.

In order to find n/16 points to discard, a great deal of cleverness is
required. The algorithm makes heavy use of two subroutines:

median(S, >)

Takes a set S of elements and a
metric for comparing them pairwise, and returns the median of the elements.

MEC-center(S, L)

Takes a set S of points and a line
L, and determines which side of L the center of the MEC of S lies on.

As mentioned above, median() predates Megiddo's work,
whereas the algorithm described here as MEC-center() was presented as part
of his 1983 paper. To explore these procedures in detail would go beyond the
scope of this outline, but each uses prune-and-search to run in linear time. The
algorithm used by MEC-center() amounts to a
simplified version of the algorithm as a whole.

Given these primitives, the algorithm for discarding n/16 input points runs
as follows:

Arbitrarily pair up the n points in S to get n/2 pairs.

Construct a bisecting line for each pair of points, to
get n/2 bisectors in total.

Call median() to find the bisector
with median slope. Call this slope mmid.

Pair up each bisector of slope ≥ mmid with
another of slope < mmid, to get n/4 intersection points. Call the
set of these points I.

Call median() to find the point in I
with median y-value. Call this y-value ymid.

Call MEC-center() to find which side
of the line y=ymid the MEC-center C lies on. (Without loss of
generality, suppose it lies above.)

Let I' be the subset of points of I whose y-values are
less than ymid. (I' contains n/8 points.)

Find a line L with slope mmid such that half
the points in I' lie to L's left, half to its right.

Let I'' be the subset of I' whose points lie to the left
of L. (I'' contains n/16 points.)

We can now discard one point in S for each of the n/16
intersection points in I''. The reasoning runs as follows. After two calls to
MEC-center(), we have found that the MEC-center C must lie
above ymid and to the right of L, whereas any point in I'' is below ymid
and to the left of L.

Each point in I'' is at the meeting point of two bisector lines. One of these
bisectors must have slope ≥ mmid, and therefore must never pass
through the quadrant where we know C to lie. Call this bisector B. Now, we know
which side of B C lies on, so of the two points whose bisector is B, let PC
be the one that lies on the same side as C, and let the other be PX.

It is easy to show that PC must be nearer to C than PX.
It follows that PC cannot lie on the minimal enclosing circle, and
thus we can safely discard a point PC for each of the n/16
intersection points in I''.

We have not discussed here how this algorithm can be made to deal with
degenerate input (parallel bisectors, collinear points, etc), but it turns out
that we get the same performance guarantees for such cases. The fact of the
matter is for degenerate input the algorithm is able to discard more than n/16
points. In short, Megiddo's algorithm guarantees to prune at least n/16 points
in each iteration independent to input.

Therefore, by the argument based on the geometric series, Megiddo's algorithm
computes the minimal enclosing circle in linear time.